Natural Deduction with Alternatives Greg Restall applied proof - - PowerPoint PPT Presentation

Natural Deduction with Alternatives

Greg Restall

applied proof theory workshop · 6 november 2020

https://consequently.org/presentation/2020/ natural-deduction-with-alternatives

My Aim To introduce naturaldeductionwithalternatives, a well-behaved single-conclusion natural deduction framework for a range of logical systems, including classical, linear, relevant logic and affine logic, by varying the policy for managing discharging of assumptions and retrieval of alternatives.

Greg Restall Natural Deduction, with Alternatives 2 of 72

My Plan

Inferentialism & Natural Deduction Classical Sequent Calculus Assertion, Denial, Negation and Contradiction Alternatives Normalisation and its Consequences Operational Rules as Definitions

Greg Restall Natural Deduction, with Alternatives 3 of 72

inferentialism & natural deduction

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

Natural Deduction is Beautiful!

[¬(r ∨ s)]4 p → (q ∨ r) [p]3

→E

q ∨ r q → s [q]1

→E

s

∨I

r ∨ s [r]2

∨I

r ∨ s

∨E1,2

r ∨ s

¬E

⊥

¬I3

¬p

→I4

¬(r ∨ s) → ¬p

Greg Restall Natural Deduction, with Alternatives 5 of 72

The Rules

A

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A

∨I

A ∨ B Π B

∨I

A ∨ B

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A

∨I

A ∨ B Π B

∨I

A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A

∨I

A ∨ B Π B

∨I

A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C [A]i Π ⊥

¬Ii

¬A

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A

∨I

A ∨ B Π B

∨I

A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C [A]i Π ⊥

¬Ii

¬A Π ¬A Π′ A ¬E ⊥

Greg Restall Natural Deduction, with Alternatives 6 of 72

The Rules

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A

∨I

A ∨ B Π B

∨I

A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C [A]i Π ⊥

¬Ii

¬A Π ¬A Π′ A ¬E ⊥ Π ⊥ ⊥E A

Greg Restall Natural Deduction, with Alternatives 6 of 72

What makes natural deduction naturaldeduction? ◮ Proofs are direct, from premise(s) to conclusion(s).

Greg Restall Natural Deduction, with Alternatives 7 of 72

What makes natural deduction naturaldeduction? ◮ Proofs are direct, from premise(s) to conclusion(s). ◮ Proofs are structures made out of formulas.

Greg Restall Natural Deduction, with Alternatives 7 of 72

What makes natural deduction naturaldeduction? ◮ Proofs are direct, from premise(s) to conclusion(s). ◮ Proofs are structures made out of formulas. ◮ Te inferential relationships between those formulas is implicit in the structure of the proof.

Greg Restall Natural Deduction, with Alternatives 7 of 72

What makes natural deduction naturaldeduction? ◮ Proofs are direct, from premise(s) to conclusion(s). ◮ Proofs are structures made out of formulas. ◮ Te inferential relationships between those formulas is implicit in the structure of the proof. ◮ Rules for the connectives are, typically, separable.

Greg Restall Natural Deduction, with Alternatives 7 of 72

What makes natural deduction naturaldeduction? ◮ Proofs are direct, from premise(s) to conclusion(s). ◮ Proofs are structures made out of formulas. ◮ Te inferential relationships between those formulas is implicit in the structure of the proof. ◮ Rules for the connectives are, typically, separable. ◮ Proofs normalise. (We can straighten out detours.)

Greg Restall Natural Deduction, with Alternatives 7 of 72

Normalisation

[A]i Π1 B

→Ii

A → B Π2 A

→E

B

Greg Restall Natural Deduction, with Alternatives 8 of 72

Normalisation

[A]i Π1 B

→Ii

A → B Π2 A

→E

B

• Π2

A Π1 B

Greg Restall Natural Deduction, with Alternatives 8 of 72

Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn.

Greg Restall Natural Deduction, with Alternatives 9 of 72

Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn. ◮ A proof from X to A shows show to meet a justification request for A against a background of granting X.

Greg Restall Natural Deduction, with Alternatives 9 of 72

Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn. ◮ A proof from X to A shows show to meet a justification request for A against a background of granting X. ◮ I/E rules play a similar role to truth conditions.

Greg Restall Natural Deduction, with Alternatives 9 of 72

Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn. ◮ A proof from X to A shows show to meet a justification request for A against a background of granting X. ◮ I/E rules play a similar role to truth conditions. ◮ Normal proofs are analytic.

Greg Restall Natural Deduction, with Alternatives 9 of 72

Natural Deduction and the Sequent Calculus ◮ Sequents (like X A) are a good way to ‘keep score.’

Greg Restall Natural Deduction, with Alternatives 10 of 72

Natural Deduction and the Sequent Calculus ◮ Sequents (like X A) are a good way to ‘keep score.’ ◮ Structural rules, like identity, cut, contraction and weakening, are typically explicit in a sequent system and implicit in natural deduction.

Greg Restall Natural Deduction, with Alternatives 10 of 72

Structural Rules A

Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules A A A

Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules Y X Π1 A Π2 B X A Y, A B X, Y B

Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules Y X Π1 A Π2 B X A Y, A B X, Y B

Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules Y X Π1 A Π2 B X A Y, A B X, Y B

Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X [A]i [A]i Π B

→Ii

A → B A

→E

B X, A, A B X, A B

Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X [A]i [A]i Π B

→Ii

A → B A

→E

B X, A, A B X, A B

Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X Π B

→I

A → B A

→E

B X B X, A B

Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X Π B

→I

A → B A

→E

B X B X, A B

Greg Restall Natural Deduction, with Alternatives 11 of 72

Discharge Policies duplicates no duplicates vacuous Standard Affine no vacuous Relevant Linear

Greg Restall Natural Deduction, with Alternatives 12 of 72

Natural deduction is opinionated ⊢ p ∨ ¬p ¬¬p ⊢ p

Greg Restall Natural Deduction, with Alternatives 13 of 72

Natural deduction is opinionated ⊢ p ∨ ¬p ¬¬p ⊢ p ⊢ (p → q) ∨ (q → r) ⊢ (((p → q)) → p) → p

Greg Restall Natural Deduction, with Alternatives 13 of 72

‘Textbook’ natural deduction plugs the gap, but it has no taste. Π ¬¬A

DNE

A [¬A]i Π ⊥

⊥Ec

A [A]i Π C [¬A]j Π C

Casesi,j

C

Greg Restall Natural Deduction, with Alternatives 14 of 72

We get classicallogic, but the rules are no longer separated

[¬p]2 [(p → q) → p]3 [¬p]2 [p]1

¬E

⊥ ⊥E q

→I1

p → q

→E

p

¬E

⊥

¬I2

¬¬p

DNE

p

→I3

((p → q) → p) → p

Greg Restall Natural Deduction, with Alternatives 15 of 72

classical sequent calculus

Gentzen’s Sequent Calculus

p p p q, p

→R

p → q, p

→L

(p → q) → p p, p

W

(p → q) → p p

→R

((p → q) → p) → p

Greg Restall Natural Deduction, with Alternatives 17 of 72

Gentzen’s Sequent Calculus

p p p q, p

→R

p → q, p

→L

(p → q) → p p, p

W

(p → q) → p p

→R

((p → q) → p) → p p p

¬R

p, ¬p

∨R

p ∨ ¬p p p

¬L

p, ¬p

∧L

p ∧ ¬p

Greg Restall Natural Deduction, with Alternatives 17 of 72

Gentzen’s Sequent Calculus

p p p q, p

→R

p → q, p

→L

(p → q) → p p, p

W

(p → q) → p p

→R

((p → q) → p) → p p p

¬R

p, ¬p

∨R

p ∨ ¬p p p

¬L

p, ¬p

∧L

p ∧ ¬p Classical • Separated Rules • Normalising • Analytic

Greg Restall Natural Deduction, with Alternatives 17 of 72

Gentzen’s Sequent Calculus

p p p q, p

→R

p → q, p

→L

(p → q) → p p, p

W

(p → q) → p p

→R

((p → q) → p) → p p p

¬R

p, ¬p

∨R

p ∨ ¬p p p

¬L

p, ¬p

∧L

p ∧ ¬p Classical • Separated Rules • Normalising • Analytic ... but what kind of proof does X Y score?

Greg Restall Natural Deduction, with Alternatives 17 of 72

Me, in 2005: Not a proof, but . . .

https://consequently.org/writing/multipleconclusions/

. . . deriving X Y does tell you that it’s out of bounds to assert each member of X and deny each member of Y, and that’s something!

Greg Restall Natural Deduction, with Alternatives 18 of 72

Steinberger on the Principle of Answerability

Florian Steinberger, “Why Conclusions Should Remain Single” JPL (2011) 40:333–355 https://dx.doi.org/10.1007/s10992-010-9153-3

Greg Restall Natural Deduction, with Alternatives 19 of 72

This is not just conservatism What is a proof of p?

Greg Restall Natural Deduction, with Alternatives 20 of 72

This is not just conservatism What is a proof of p? A proof of p meets a justification request for the assertion of p.

(Not every way to meet a justification request is a proof, but proofs meet justification requests in a very stringent way.)

Greg Restall Natural Deduction, with Alternatives 20 of 72

Slogan A proof of A (in a context) meets a justification request for A

• n the basis of the claims we take for granted.

Greg Restall Natural Deduction, with Alternatives 21 of 72

Slogan A proof of A (in a context) meets a justification request for A

• n the basis of the claims we take for granted.

A sequent calculus derivation doesn’t do that, at least, not without quite a bit of work.

Greg Restall Natural Deduction, with Alternatives 21 of 72

Slogan A proof of A (in a context) meets a justification request for A

• n the basis of the claims we take for granted.

A sequent calculus derivation doesn’t do that, at least, not without quite a bit of work. Is there a way to read a classical sequent derivation as constructing this kind of proof?

Greg Restall Natural Deduction, with Alternatives 21 of 72

Signed Natural Deduction [− p ∨ ¬p]1

−∨E

− p

+¬I

+ ¬p

+∨I

+ p ∨ ¬p [− p ∨ ¬p]2

RAA1,2

+ p ∨ ¬p Decorate your proof with signs.

Greg Restall Natural Deduction, with Alternatives 22 of 72

Double up your Rules

Π + A

+∨I

+ A ∨ B Π + B

+∨I

+ A ∨ B Π + A ∨ B [+ A]j Π′ φ [+ B]k Π′′ φ

∨Ej,k

φ Π − A ∨ B −∨E − A Π − A ∨ B −∧E − B Π − A Π′ − B −∨E − A ∨ B Π − A

+¬I

+ ¬A Π + ¬A +¬E − A Π + A

−¬I

− ¬A Π − ¬A −¬E + A

Greg Restall Natural Deduction, with Alternatives 23 of 72

Add some ‘Structural’ Rules

Π α Π′ α∗

⊥I

⊥ [α]i Π ⊥ Reductioi α∗ [α]j Π′ β [α]k Π′′ β∗

SRj,k

α∗

α and β are signed formulas. (− A)∗ = + A and (+ A)∗ = − A.

Greg Restall Natural Deduction, with Alternatives 24 of 72

This is very complex Te duality of assertion and denial are important to the defender of classical logic, but doubling up every connective rule is like cracking a small nut with a sledgehammer.

Greg Restall Natural Deduction, with Alternatives 25 of 72

So far ... ◮ Answerability to our practice is a constraint worth meeting.

Greg Restall Natural Deduction, with Alternatives 26 of 72

So far ... ◮ Answerability to our practice is a constraint worth meeting. ◮ Sequents help keep score in a proof.

Greg Restall Natural Deduction, with Alternatives 26 of 72

So far ... ◮ Answerability to our practice is a constraint worth meeting. ◮ Sequents help keep score in a proof. ◮ Bilateralism (paying attention to assertion and denial) is important to the defender of classical logic.

Greg Restall Natural Deduction, with Alternatives 26 of 72

So far ... ◮ Answerability to our practice is a constraint worth meeting. ◮ Sequents help keep score in a proof. ◮ Bilateralism (paying attention to assertion and denial) is important to the defender of classical logic. ◮ Sequent calculus and signed natural deduction do not approach the simplicity of standard natural deduction as an account of proof.

Greg Restall Natural Deduction, with Alternatives 26 of 72

assertion, denial, negation and contradiction

Are these rules truly separated?

[A]i Π ⊥

¬Ii

¬A Π ¬A Π′ A ¬E ⊥ Π ⊥ ⊥E A

Greg Restall Natural Deduction, with Alternatives 28 of 72

Are these rules truly separated?

[A]i Π ⊥

¬Ii

¬A Π ¬A Π′ A ¬E ⊥ Π ⊥ ⊥E A

If ⊥ is a formula we do not have the subformula property for normal proofs. ¬p p

¬E

⊥ ⊥E q

Greg Restall Natural Deduction, with Alternatives 28 of 72

In the sequent calculus, it’s structure, not a formula

¬p p

¬E

⊥ ⊥E q p p

¬L

¬p, p

K

¬p, p q

Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure, not a formula

¬p p

¬E

⊥ ⊥E q p p

¬L

¬p, p

K

¬p, p q

Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure, not a formula

¬p p

¬E

⊥ ⊥E q p p

¬L

¬p, p

K

¬p, p q

Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure, not a formula

¬p p

¬E

⊥ ⊥E q p p

¬L

¬p, p

K

¬p, p q

Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure, not a formula

¬p p

¬E

♯

♯E

q p p

¬L

¬p, p

K

¬p, p q

Following Tennant (Natural Logic, 1978), I’ll use “♯” as a contradiction marker. It’s not a formula. We regain the subformula property for normal proofs.

Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure, not a formula

¬p p

¬E

♯

♯E

q p p

¬L

¬p, p

K

¬p, p q

Following Tennant (Natural Logic, 1978), I’ll use “♯” as a contradiction marker. It’s not a formula. We regain the subformula property for normal proofs. [A]i Π ♯

¬Ii

¬A Π ¬A Π′ A ¬E ♯ Π ♯

♯E

A

Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure, not a formula

¬p p

¬E

♯

♯E

q p p

¬L

¬p, p

K

¬p, p q

Following Tennant (Natural Logic, 1978), I’ll use “♯” as a contradiction marker. It’s not a formula. We regain the subformula property for normal proofs. [A]i Π ♯

¬Ii

¬A Π ¬A Π′ A ¬E ♯ Π ♯

♯E

A Π ♯

fI

f Π f fE ♯

Greg Restall Natural Deduction, with Alternatives 29 of 72

Relevance, Vacuous Discharge, and ♯E

¬p p

¬E

♯

♯E

q p p

¬L

¬p, p

K

¬p, p q

Greg Restall Natural Deduction, with Alternatives 30 of 72

Relevance, Vacuous Discharge, and ♯E

¬p p

¬E

♯

♯E

q p p

¬L

¬p, p

K

¬p, p q

p

→I

q → p p p

K

p, q p

→I

p q → p

Greg Restall Natural Deduction, with Alternatives 30 of 72

Relevance, Vacuous Discharge, and ♯E

¬p p

¬E

♯

♯E

q p p

¬L

¬p, p

K

¬p, p q

p

→I

q → p p p

K

p, q p

→I

p q → p

Greg Restall Natural Deduction, with Alternatives 30 of 72

Relevance, Vacuous Discharge, and ♯E

¬p p

¬E

♯

♯E

q p p

¬L

¬p, p

K

¬p, p q

p

→I

q → p p p

K

p, q p

→I

p q → p

Greg Restall Natural Deduction, with Alternatives 30 of 72

Relevance, Vacuous Discharge, and ♯E

¬p p

¬E

♯

♯E

q p p

¬L

¬p, p

K

¬p, p q

p

→I

q → p p p

K

p, q p

→I

p q → p What connects vacuous discharge and ♯E? In the sequent calculus, they are both weakening. But in natural deduction?

Greg Restall Natural Deduction, with Alternatives 30 of 72

Is ♯ genuinely structural? If ♯ is a genuine structural feature of proofs, why does it feature only in the f and ¬ rules?

Greg Restall Natural Deduction, with Alternatives 31 of 72

Is ♯ genuinely structural? If ♯ is a genuine structural feature of proofs, why does it feature only in the f and ¬ rules? For bilateralists, the notion of a contradiction is more fundamental than any particular connective. Asserting and denying the one thing can also lead to a dead end ...

Greg Restall Natural Deduction, with Alternatives 31 of 72

Is ♯ genuinely structural? If ♯ is a genuine structural feature of proofs, why does it feature only in the f and ¬ rules? For bilateralists, the notion of a contradiction is more fundamental than any particular connective. Asserting and denying the one thing can also lead to a dead end ... and so can setting aside the current conclusion, to look for an alternative.

Greg Restall Natural Deduction, with Alternatives 31 of 72

alternatives

Adding alternatives Π A A↑

↑

♯

Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A

Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A

[X : Y] A

↑

[X : A, Y] ♯

Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A

[X : Y] A

↑

[X : A, Y] ♯

Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A

[X : Y] A

↑

[X : A, Y] ♯ [X : A, Y] ♯

↓

[X : Y] A

Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A

[X : Y] A

↑

[X : A, Y] ♯ [X : A, Y] ♯

↓

[X : Y] A

Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A

X A; Y

↑

X ♯; A, Y X ♯; A, Y

↓

X A; Y

Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A

X A; Y

↑

X ♯; A, Y X ♯; A, Y

↓

X A; Y

We add the store and retrieve rules and keep the other rules fixed. Te store and retrieve rules are the only rules that manipulate alternatives.

Greg Restall Natural Deduction, with Alternatives 33 of 72

A minimal set of rules

A Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π ♯

fI

f Π f fE ♯

Greg Restall Natural Deduction, with Alternatives 34 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[A : ] A

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[A : A] ♯

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[A : A] f

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[ : A] A → f

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[(A → f) → B : A] B

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[B → f, (A → f) → B : A] f

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[B → f, (A → f) → B : A] ♯

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[B → f, (A → f) → B : ] A

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition

[(A → f) → B : ] (B → f) → A

[B → f]3 (A → f) → B [A]1 [A↑]2

↓

♯

fI

f

→I1

A → f

→E

B

→E

f fE ♯

↓2

A

→I3

(B → f) → A

[(A → f) → B : ] (B → f) → A

Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Peirce’s Law

[(p → q) → p : ] p (p → q) → p [p]1 [p↑]2

↑

♯

↓

q

→I1

p → q

→E

p [p↑]2

↑

♯

↓2

p

Greg Restall Natural Deduction, with Alternatives 36 of 72

Example Proof: Peirce’s Law

[(p → q) → p : ] p (p → q) → p [p]1 [p↑]2

↑

♯

↓

q

→I1

p → q

→E

p [p↑]2

↑

♯

↓2

p Tis proof exhibits both duplicate and vacuous retrieval.

Greg Restall Natural Deduction, with Alternatives 36 of 72

Example Proof: Peirce’s Law

[(p → q) → p : ] p (p → q) → p [p]1 [p↑]2

↑

♯

↓

q

→I1

p → q

→E

p [p↑]2

↑

♯

↓2

p Tis proof exhibits both duplicate and vacuous retrieval.

Greg Restall Natural Deduction, with Alternatives 36 of 72

The Unity of Relevance ¬p p

¬E

♯

♯E

q p

→I

q → p

Greg Restall Natural Deduction, with Alternatives 37 of 72

The Unity of Relevance ♯E is nothing other than vacuous retrieval. ¬p p

¬E

♯

↓

q p

→I

q → p

Greg Restall Natural Deduction, with Alternatives 37 of 72

Completeness and Soundness — for classical logic

• 1. completeness:

Greg Restall Natural Deduction, with Alternatives 38 of 72

Completeness and Soundness — for classical logic

• 1. completeness: Trivial. It’s intuitionistic logic + Peirce’s Law

Greg Restall Natural Deduction, with Alternatives 38 of 72

Completeness and Soundness — for classical logic

• 1. completeness: Trivial. It’s intuitionistic logic + Peirce’s Law
• 2. soundness: Easy induction. If we have a proof for [X : Y] A then

in any Boolean valuation v where v(X) = 1 and v(Y) = 0 then v(A) = 1.

Greg Restall Natural Deduction, with Alternatives 38 of 72

Bilateralism does some work for us When I set a current conclusion aside as an alternative, I temporarily (for the sake of the argument) deny it, to consider a different option in its place. Tis is very mildly bilateral, but not so much that it litters every formula in a proof with a sign.

Greg Restall Natural Deduction, with Alternatives 39 of 72

Benefits Classical (and Linear, Relevant, and Affine, too) Separated Rules • Normalising Analytic • Single Conclusion • Answerable

Greg Restall Natural Deduction, with Alternatives 40 of 72

normalisation and its consequences

Flattening Local Peaks: →I/→E

[A]i Π1 B

→Ii

A → B Π2 A

→E

B

Greg Restall Natural Deduction, with Alternatives 42 of 72

Flattening Local Peaks: →I/→E

[A]i Π1 B

→Ii

A → B Π2 A

→E

B →I/→E Π2 A Π1 B

Greg Restall Natural Deduction, with Alternatives 42 of 72

Flattening Local Peaks: →I/→E

[A]i Π1 B

→Ii

A → B Π2 A

→E

B →I/→E Π2 A Π1 B

note: if the original proof satisfies a given discharge policy, so does its reduction. To show this we need to ensure that duplicate/vacuous discharge is banned whenever duplicate/vacuous retrieval is banned.

Greg Restall Natural Deduction, with Alternatives 42 of 72

Flattening Local Peaks: fI/fE

Π ♯

fI

f

fE

♯

Greg Restall Natural Deduction, with Alternatives 43 of 72

Flattening Local Peaks: fI/fE

Π ♯

fI

f

fE

♯ fI/fE Π ♯

Greg Restall Natural Deduction, with Alternatives 43 of 72

Flattening Local Peaks: ↓/↑

[A↑]i Π ♯

↓i

A A↑

↓

♯

Greg Restall Natural Deduction, with Alternatives 44 of 72

Flattening Local Peaks: ↓/↑

[A↑]i Π ♯

↓i

A A↑

↓

♯ ↓/↑ A↑ Π ♯

Greg Restall Natural Deduction, with Alternatives 44 of 72

One more case to consider ... What about →I/↑/↓/→E sequences?

Greg Restall Natural Deduction, with Alternatives 45 of 72

Flattening Local Peaks: ↓/→E

[A → B↑]i Π1 ♯

↓i

A → B Π2 A

→E

B

Greg Restall Natural Deduction, with Alternatives 46 of 72

Flattening Local Peaks: ↓/→E

[A → B↑]i Π1 ♯

↓i

A → B Π2 A

→E

B ↓/→E [B↑]j Π∗

1

♯

↓j

B

Where Π∗

1 is the proof:

Π1      . . . A → B A → B↑

↑i

♯ := . . . A → B Π2 A →E B B↑

↑j

♯     

Greg Restall Natural Deduction, with Alternatives 46 of 72

Flattening Local Peaks: ↓/→E

[A → B↑]i Π1 ♯

↓i

A → B Π2 A

→E

B ↓/→E [B↑]j Π∗

1

♯

↓j

B

Where Π∗

1 is the proof:

Π1      . . . A → B A → B↑

↑i

♯ := . . . A → B Π2 A →E B B↑

↑j

♯     

note: if the original proof satisfies a given discharge policy, so does its reduction. To show this we need to ensure that duplicate/vacuous retrieval is banned whenever duplicate/vacuous discharge is banned.

Greg Restall Natural Deduction, with Alternatives 46 of 72

Normalisation and Strong Normalisation

It’s straightforward to show that any proof Π can be transformed, in some finite series of reduction steps, into a proof Π′, to which no reduction applies. Such a proof is normal.

Greg Restall Natural Deduction, with Alternatives 47 of 72

Normalisation and Strong Normalisation

It’s straightforward to show that any proof Π can be transformed, in some finite series of reduction steps, into a proof Π′, to which no reduction applies. Such a proof is normal. It’s less straightforward to show that any sequence

• f reduction steps applied to a proof Π will terminate,

after finitely many steps. (Parigot JSL 1997)

Greg Restall Natural Deduction, with Alternatives 47 of 72

Normalisation and Strong Normalisation

It’s straightforward to show that any proof Π can be transformed, in some finite series of reduction steps, into a proof Π′, to which no reduction applies. Such a proof is normal. It’s less straightforward to show that any sequence

• f reduction steps applied to a proof Π will terminate,

after finitely many steps. (Parigot JSL 1997) Normal proofs are analytic. Every formula in a normal proof from [X : Y] to A is a subformula of sone formula in X, Y or A.

Greg Restall Natural Deduction, with Alternatives 47 of 72

Further reductions: →E/→I

Π A → B [A]i

→E

B

→Ii

A → B

Greg Restall Natural Deduction, with Alternatives 48 of 72

Further reductions: →E/→I

Π A → B [A]i

→E

B

→Ii

A → B →E/→I Π A → B

Greg Restall Natural Deduction, with Alternatives 48 of 72

Further reductions: fE/fI

Π f

fE

♯

fI

f

Greg Restall Natural Deduction, with Alternatives 49 of 72

Further reductions: fE/fI

Π f

fE

♯

fI

f fE/fI Π f

Greg Restall Natural Deduction, with Alternatives 49 of 72

Further reductions: ↑/↓

Π A [A↑]i

↑

♯

↓i

A

Greg Restall Natural Deduction, with Alternatives 50 of 72

Further reductions: ↑/↓

Π A [A↑]i

↑

♯

↓i

A ↑/↓ Π A

Greg Restall Natural Deduction, with Alternatives 50 of 72

The rules

A Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π ♯

fI

f Π f fE ♯

Greg Restall Natural Deduction, with Alternatives 51 of 72

Defining Negation: ¬A =df A → f

A → f A →E f fE ♯ ¬A A ¬E ♯

Greg Restall Natural Deduction, with Alternatives 52 of 72

Defining Negation: ¬A =df A → f

A → f A →E f fE ♯ ¬A A ¬E ♯ [A]i Π ♯

fI

f

→Ii

A → f [A]i Π ♯

¬Ii

¬A

Greg Restall Natural Deduction, with Alternatives 52 of 72

Negation Reductions: ¬I/¬E

[A]i Π1 ♯

¬Ii

¬A Π2 A

¬E

♯ ¬I/¬E Π2 A Π1 ♯

Greg Restall Natural Deduction, with Alternatives 53 of 72

Negation Reductions: ¬I/¬E

[A]i Π1 ♯

¬Ii

¬A Π2 A

¬E

♯ ¬I/¬E Π2 A Π1 ♯ [A]i Π1 ♯

fI

f

→Ii

A → f Π2 A

→E

f fE ♯

Greg Restall Natural Deduction, with Alternatives 53 of 72

Negation Reductions: ¬I/¬E

[A]i Π1 ♯

¬Ii

¬A Π2 A

¬E

♯ ¬I/¬E Π2 A Π1 ♯ [A]i Π1 ♯

fI

f

→Ii

A → f Π2 A

→E

f fE ♯ →I/→E Π2 A Π1 ♯

fI

f

fE

♯

Greg Restall Natural Deduction, with Alternatives 53 of 72

Negation Reductions: ¬I/¬E

[A]i Π1 ♯

¬Ii

¬A Π2 A

¬E

♯ ¬I/¬E Π2 A Π1 ♯ [A]i Π1 ♯

fI

f

→Ii

A → f Π2 A

→E

f fE ♯ →I/→E Π2 A Π1 ♯

fI

f

fE

♯ fI/fE Π2 A Π1 ♯

Greg Restall Natural Deduction, with Alternatives 53 of 72

Defining Disjunction: A ⊕ B =df ¬A → B

¬A → B [A]1 A↑

↑

♯

¬I1

¬A →E B A ⊕ B A↑

⊕E

B

Greg Restall Natural Deduction, with Alternatives 54 of 72

Defining Disjunction: A ⊕ B =df ¬A → B

¬A → B [A]1 A↑

↑

♯

¬I1

¬A →E B A ⊕ B A↑

⊕E

B

[¬A]3 [A↑]1 Π B [B↑]2

↑

♯

↓1

A

¬E

♯

↓2

B

→I3

¬A → B [A↑]1 Π B

⊕I1

A ⊕ B

Greg Restall Natural Deduction, with Alternatives 54 of 72

Disjunction Reductions: ⊕I/⊕E

[A↑]1 Π B

⊕I1

A ⊕ B A↑

⊕E

B ⊕I/⊕E A↑ Π B

Greg Restall Natural Deduction, with Alternatives 55 of 72

Disjunction Reductions: ⊕I/⊕E

[A↑]1 Π B

⊕I1

A ⊕ B A↑

⊕E

B ⊕I/⊕E A↑ Π B [¬A]3 [A↑]1 Π B [B↑]2

↑

♯

↓1

A

¬E

♯

↓2

B

→I3

¬A → B [A]4 A↑

↑

♯

¬I4

¬A

→E

B →I/→E

Greg Restall Natural Deduction, with Alternatives 55 of 72

Disjunction Reductions: ⊕I/⊕E

[A]4 A↑

↑

♯

¬I4

¬A [A↑]1 Π B [B↑]2

↑

♯

↓1

A

¬E

♯

↓2

B ¬I/¬E

Greg Restall Natural Deduction, with Alternatives 55 of 72

Disjunction Reductions: ⊕I/⊕E

[A↑]1 Π B [B↑]2

↑

♯

↓1

A A↑

↑

♯

↓2

B ↓/↑

Greg Restall Natural Deduction, with Alternatives 55 of 72

Disjunction Reductions: ⊕I/⊕E

A↑ Π B [B↑]2

↑

♯

↓2

B ↑/↓

Greg Restall Natural Deduction, with Alternatives 55 of 72

Disjunction Reductions: ⊕I/⊕E

A↑ Π B [B↑]2

↑

♯

↓2

B ↑/↓ A↑ Π B

Greg Restall Natural Deduction, with Alternatives 55 of 72

Defining Conjunction: A ⊗ B =df ¬(A → ¬B)

[A → ¬B]1 A

→E

¬B B ¬E ♯

¬I1

¬(A → ¬B) A B ⊗I A ⊗ B

Greg Restall Natural Deduction, with Alternatives 56 of 72

Defining Conjunction: A ⊗ B =df ¬(A → ¬B)

[A → ¬B]1 A

→E

¬B B ¬E ♯

¬I1

¬(A → ¬B) A B ⊗I A ⊗ B

¬(A → ¬B) [A]1 [B]2 Π C [C↑]3

↑

♯

¬I2

¬B

→I1

A → ¬B

¬E

♯

↓3

C A ⊗ B [A]1 [B]2 Π C ⊗E1,2 C

Greg Restall Natural Deduction, with Alternatives 56 of 72

Conjunction Reductions: ⊗I/⊗E

Π1 A Π2 B ⊗I A ⊗ B [A]1 [B]2 Π C

⊗E1,2

C ⊗I/⊗E Π1 A Π2 B Π C

Greg Restall Natural Deduction, with Alternatives 57 of 72

Conjunction Reductions: ⊗I/⊗E

Π1 A Π2 B ⊗I A ⊗ B [A]1 [B]2 Π C

⊗E1,2

C ⊗I/⊗E Π1 A Π2 B Π C

[A → ¬B]1 Π1 A →E ¬B Π2 B ¬E ♯ ¬I1 ¬(A → ¬B) [A]1 [B]2 Π C [C↑]3 ↑ ♯ ¬I2 ¬B →I1 A → ¬B ¬E ♯ ↓3 C

Greg Restall Natural Deduction, with Alternatives 57 of 72

Conjunction Reductions: ⊗I/⊗E

Π1 A Π2 B ⊗I A ⊗ B [A]1 [B]2 Π C

⊗E1,2

C ⊗I/⊗E Π1 A Π2 B Π C

[A → ¬B]1 Π1 A →E ¬B Π2 B ¬E ♯ ¬I1 ¬(A → ¬B) [A]1 [B]2 Π C [C↑]3 ↑ ♯ ¬I2 ¬B →I1 A → ¬B ¬E ♯ ↓3 C Tis reduces as desired, using ¬I/¬E , →I/→E , ¬I/¬E and ↑/↓ reductions.

Greg Restall Natural Deduction, with Alternatives 57 of 72

Reduction Steps for the full vocabulary →I/→E , fI/fE , ↓/↑ , ↓/→E and ↑/↓

Greg Restall Natural Deduction, with Alternatives 58 of 72

• perational rules

as definitions

Structural Rules and Operational Rules

A Π A A↑

↑

♯ [A↑]i Π ♯

↓i

A [A]i Π B

→Ii

A → B Π A → B Π′ A →E B Π ♯

fI

f Π f fE ♯ In what sense can the I/E rules for a concept be understood as defining it?

Greg Restall Natural Deduction, with Alternatives 60 of 72

Defining Rules

https://consequently.org/writing/generality-and-existence-1/

Greg Restall Natural Deduction, with Alternatives 61 of 72

Examples of Defining Rules

X A, B, Y

⊕Df

X A ⊕ B, Y X, A, B Y

⊗Df

X, A ⊗ B Y X A, Y

¬Df

X, ¬A Y X Y

fDf

X f, Y X, A B, Y

→Df

X A → B, Y X, A B, Y

−Df

X, A − B Y Tis gives the conditions under which an assertion [left side]

• r denial [right side] of the formula is out of bounds.

Greg Restall Natural Deduction, with Alternatives 62 of 72

Each Defining Rule (using Cut/Id) gives rise to Left/Right Rules

X A, Y

¬Df

X, ¬A Y X A, Y

¬L

X, ¬A Y X, A Y

¬R

X ¬A, Y X, A, B Y

⊗Df

X, A ⊗ B Y X, A, B Y

⊗L

X, A ⊗ B Y X A, Y X′ B, Y ′

⊗R

X, X′ A ⊗ B, Y, Y ′ X A, B, Y

⊕Df

X A ⊕ B, Y X, A Y X′, B Y ′

⊕L

X, X′, A ⊕ B Y, Y ′ X A, B, Y

⊕R

X A ⊕ B, Y

Te Left/Right rules arising in this way admit a straightforward Cut-elimination proof.

Greg Restall Natural Deduction, with Alternatives 63 of 72

Conservative Extension

Df + Cut + Id ↔ L/R + Cut + Id ↔ L/R + Id

Greg Restall Natural Deduction, with Alternatives 64 of 72

Adding Focus

X A; Y

¬FDf

X, ¬A ♯; Y X A; Y

¬FL

X, ¬A ♯; Y X, A ♯; Y

¬FR

X ¬A; Y X, A, B C; Y

⊗FDf

X, A ⊗ B C; Y X, A, B C; Y

⊗FL

X, A ⊗ B C; Y X A; Y X′ B; Y ′

⊗FR

X, X′ A ⊗ B; Y, Y ′ X A; B, Y

⊕FDf

X A ⊕ B; Y X, A ♯; Y X′, B ♯; Y ′

⊕FL

X, X′, A ⊕ B ♯; Y, Y ′ X A; B, Y

⊕FR

X A ⊕ B; Y X B; A, Y

⊕FDf ′

X A ⊕ B; Y X, A ♯; Y X′, B ♯; Y ′

⊕FL ′

X, X′, A ⊕ B ♯; Y, Y ′ X B; A, Y

⊕FR ′

X A ⊕ B; Y

Tere is more than one way to add focus to a defining rule or a pair of left/right rules.

Greg Restall Natural Deduction, with Alternatives 65 of 72

Conservative Extension

Df + Cut + Id ↔ L/R + Cut + Id ↔ L/R + Id

• FDf + FCut + FId + ↑/↓

FL/FR + FCut + FId + ↑/↓ FL/FR + FId + ↑/↓

Greg Restall Natural Deduction, with Alternatives 66 of 72

FL/FR Rules can be read as I/E rules

X A; Y

¬FL

X, ¬A ♯; Y X, A ♯; Y

¬FR

X ¬A; Y ¬A Π A ¬E ♯ [A]i Π ♯

¬Ii

¬A ... and the E rules have the major premise as an assumption. Te proofs generated from these rules (without using Cut) are normal.

Greg Restall Natural Deduction, with Alternatives 67 of 72

Conservative Extension

Df + Cut + Id ↔ L/R + Cut + Id ↔ L/R + Id

• FDf + FCut + FId + ↑/↓

FL/FR + FCut + FId + ↑/↓ FL/FR + FId + ↑/↓

• I/E Proofs

Normal I/E proofs

Greg Restall Natural Deduction, with Alternatives 68 of 72

Conservative Extension

Df + Cut + Id ↔ L/R + Cut + Id ↔ L/R + Id

• FDf + FCut + FId + ↑/↓

FL/FR + FCut + FId + ↑/↓ FL/FR + FId + ↑/↓

• I/E Proofs

Normal I/E proofs We have a systematic conservative extension result showing how any concept given by a defining rule can be given I/E rules that admit normalisation. Tis means new concepts conservatively extend the old vocabulary.

Greg Restall Natural Deduction, with Alternatives 68 of 72

Conservative Extension

Df + Cut + Id ↔ L/R + Cut + Id ↔ L/R + Id

• FDf + FCut + FId + ↑/↓

FL/FR + FCut + FId + ↑/↓ FL/FR + FId + ↑/↓

• I/E Proofs

Normal I/E proofs We have a systematic conservative extension result showing how any concept given by a defining rule can be given I/E rules that admit normalisation. Tis means new concepts conservatively extend the old vocabulary. homework: Prove this directly, without the detour through the sequent calculus.

Greg Restall Natural Deduction, with Alternatives 68 of 72

How (good) operational rules can define

Defining rules with focus settle the bounds for assertion and denial of the concepts they govern, and they also show in a systematic way how to meet justification requests for judgements involving those concepts. Tey do this in a way that this conservative and uniquely defining. What more could you want?

Greg Restall Natural Deduction, with Alternatives 69 of 72

thank you!

References and Further Reading

◮ Dag Prawitz, Natural Deduction, Almqvist and Wiksell, 1965. ◮ Neil Tennant, Natural Logic, Edinburgh University Press, 1978. ◮ Michel Parigot, “λµ-Calculus: An Algorithmic Interpretation of Classical Natural Deduction,” pp. 190–201 in International Conference on Logic for Programming Artificial Intelligence and Reasoning, edited by Andrei Voronkov, Springer Lecture Notes in Artificial Intelligence, 1992. ◮ Michel Parigot, “Proofs of Strong Normalisation for Second Order Classical Natural Deduction,” Journal of Symbolic Logic, 1997 (62), 1461–1479. ◮ Greg Restall, “Multiple Conclusions,” pp. 189–205 in Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, edited by Petr Hájek, Luis Valdés-Villanueva and Dag Westerståhl, KCL Publications, 2005. ◮ Florian Steinberger, “Why Conclusions Should Remain Single” Journal of Philosophical Logic, 2011 (40) 333–355. ◮ Nils Kürbis, Proof and Falsity, Cambridge University Press, 2019. ◮ Greg Restall, “Generality and Existence 1: Quantification and Free Logic”, Review of Symbolic Logic, 2019 (12), 1–29.

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